Starting Points: Building Story Problems

I was reading a blog post recently about how 90% of students dislike story problems. Parents ask me about this in email and it came up in my old Facebook group. What do we do about story problems and how do we learn to solve them?

Gattegno says only awareness is educable in humans. The longer I spend time with Gattegno and noticing and wondering in my own life, the more I believe he is correct. Awareness is the place we always need to start. What must my student be aware of to solve story problems?

AWARENESS AND STORY PROBLEMS

Well, story problems are models of real world problems. Rarely, in real life, do we have numbers with an operation sign attached to it. Instead, we have recipes that need to be converted, traffic that needs to be routed, bridges that need built and finances that need to be managed.

Real life problems are messy, often there is extra information and we must learn to make decisions about what information is necessary and what isn't. That isn't a simple process and we make a lot of mistakes.

That means we must be able to choose the correct information out of all the bits that are available to us. We look for relationships and connections. Sometimes, we don't have all the information we need. Sometimes we have too much information. Sometimes we aren't sure what questions we should even be asking.

Developing competent and confident problems solvers doesn't happen overnight. Students need to learn the habit of careful observation, attending to details, and understanding what the problem is asking in order to sort necessary from unnecessary information.

In Madeleine Goutard's book, Mathematics and Children, she develops student's problem solving abilities not by giving them word problems, but by providing the opportunity to for students to build their own. This requires a lot more thinking and careful judgment on the part of the student.

We call these Starting Points and they allow student's the learn to craft their own story problems long before they start solving the problems written by others. The child must make decisions about what information to include and what to exclude, what relationships do they want to talk about, how will they express those relationships using operations and fractions, and how will they present their story and problem in such a way that others understand what is being communicated and asked? Going through this process forces a host of awarenesses on a child.

Students don't learn how to use math notation unless they use it with the ideas generated from their own minds. If everything is packaged for them on a worksheet they rarely have need to think about how to use the operations to communicate information or request information.

Goutard is a proponent of students generating all the math that they do. They write their own problems and they solve their own problems. The process of writing math story problems produces the kind of awareness in a student that would be impossible to achieve from solving problems only.

SO WHERE DO WE START BUILDING STORY PROBLEMS?

When we come to a Starting Point, we want to spend some time noticing and wondering about the situation. I like to create images for us to look at like the one above. Once we have spent ample time on notice and wonder (ample is until no one has anything left to say), I have my students tell me a story about the image and what they think is going on. I ask them to do this because they need to have a story when it comes to writing the story problems. If we skip this step, the kids will flounder when it comes time to write their word problems.

EXAMINING THE BALL STORY IMAGE

Let's start with noticing. Here are a few of the things we noticed:

Those kids have a lot of balls.

There is a boy and a girl.

They are outside and it's day.

There is one tree.

There are 11 footballs.

There are 8 baseballs.

If we counted right, there are 37 balls total.

There are 4 different kinds of balls.

The girl has 4 baseballs.

We could go on with this for quite awhile. But I'm going to stop there so this blog post remains a reasonable length. Notice that some of the stuff we noticed didn't have anything to do with math. That is a good thing. In life, this is how problems work. Here were the kid's wonderings:

Where did all those balls come from?

Why are they at the park or field?

Are there any more kids?

Where are their parents?

Why didn't the kids split the balls up evenly?

I wonder if the girl would give all the footballs to the boy.

Now we want to ask the students to use their noticing and wondering to tell a story about this image. Once they've told their story, can they write story problems that fit with their story. At first, students might need help and you can do this in groups or as a family. Besides, that's more fun anyway.

How many story problems could be written that would fit the story you created? Write those down. We take turns on a Starting Point like this one. We all take turns sharing what we notice. Then we all go around the table and share what we think is going on. Here's our working story:

A tornado came and blew the roof off the school and all the balls went bouncing to the park. Bill and Becky each gathered a pile of balls.

LET'S WRITE SOME STORY PROBLEMS

A tornado hit the school gym and tore off the roof. A bunch of balls flew out. Bill and Becky decided to go gather them up. There were 100 balls in the gym, Bill gathered 21 and Becky gathered 18. How many are still in the gym?

What percent of the total number balls did Bill gather?

Bill picked up more balls than Becky. What is the fraction that represents how many more balls he gathered than Becky?

I had questions about the first story problem. The story problem says that there were 100 balls in the gym. Is that before or after the tornado? I think context tells me the answer to this, but that sentence isn't clear. This is a great opportunity to talk about my failings as a teacher to be clear by what I mean and how difficult it is to actually communicate information to other people. We all have to agree to the meaning of words and be clear in our speech.

I bet we could write 30 story problems for this image and the tornado story. If we change the story about what happened, we could write a bunch more story problems. Maybe there were 125 balls before the tornado, there are 100 left in the school, and it took them an hour to gather these balls. What if we added more kids to the story? What if there wasn't a tornado, but they were planning a field day for their school? This is an important exercise by itself. There are facts to this story, but they have meaning in context.

Noticing and wondering helps develop student's problem solving ability. Here is are some the general awarenesses students should come to understand as a result of working with Starting Points:

Not everything we notice is relevant to what we are doing.

The information you use depends on what you want to know.

Organizing information makes it easier to use later.

edit

Everyone must agree what words mean or it is impossible to communicate accurately.

edit

Context is everything.

That last awareness is a huge one. Things have meaning in a context. Total doesn't mean anything unless I know 'of what': total number of balls at the school before the storm, total number that blew out, total that Becky has, total that Bill has.

One of the things I want to give my students is the ability to encounter a problem and feel confident that using the tools they have, they can address the situation or solve the problem.

I am posting this Starting Point over at my online learning platform on Learning Well At Home. If you'd like to get a pdf copy you can get it here. Be sure to share with us your noticing and wonderings. We'd love to hear your stories for this Starting Point as well as your story problems.